A 
-automatic
sequence is a sequence whose 
th term is produced by a deterministic finite
automaton reading the base-
digits of 
, where 
is an integer with 
. A fixed output value is assigned to each state, and
the 
th
term is the output assigned to the state reached after all digits of 
have been read. Equivalently, a sequence
with values in a finite alphabet
is 
-automatic
when its 
-kernel,
formed from the subsequences 
with integers 
and 
, is finite. The Thue-Morse sequence is 2-automatic.
Automatic Sequence
See also
Automata Theory, Automatic Set, Finite Automaton, Integer Sequence, Thue-Morse SequenceExplore with Wolfram|Alpha
References
Allouche, J.-P. and Shallit, J. Automatic Sequences: Theory, Applications, Generalizations. Cambridge, England: Cambridge University Press, 2003.Cite this as:
Weisstein, Eric W. "Automatic Sequence." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AutomaticSequence.html